I have a set of points whose positions are represented by 3D vectors. Each point has the associated weight in range from 0 to 1. The sum of all weights doesn't equal to 1.
How should the weighted mean point be calculated from the whole set of points?
The process is called normalization, and you simly divide each weight by the sum of all the weights: $$w_i \rightarrow \frac{w_i}{\sum w_i}$$ You can easily verify that the sum of the new weights is now $1$.
The mean vector is given by the sum: $${\bf{v}} = \sum w_i \bf{v}_i$$